Part (a)
Knowing that acceleration is equal to the gravitation acceleration, I deduced that: d²x/dt² = -g, then I integerated to find dx/dt = -gt + C.
At t=0, (dx/dt)(0) = C .: C = v0 or initial velocity.
I'm not too certain what the question for part (a) is asking since I can't imagine a ball rising once it is released...unless it is thrown upwards?
In any case, I integrated dx/dt and got x(t) = (-1/2)gt² + (v0)t + x0, then substituted values g = 9.8 m/s², v0 = 50 ft/s = 15.24 m/s,
and x0 = 100 ft = 30.48m into the equation and got: x(t) = -4.9t² + 15.24t + 30.48
but not sure if this answers part (a).

Part (b)
Assuming that the above equation for distance that I derived is correct, I simply needed to solve for t in x(t) = 0.
I used the quadratic formula and got t = 4.49s (ignoring the negative solution since time must be a positive value).

Part (a)
Knowing that acceleration is equal to the gravitation acceleration, I deduced that: d²x/dt² = -g, then I integerated to find dx/dt = -gt + C.
At t=0, (dx/dt)(0) = C .: C = v0 or initial velocity.
I'm not too certain what the question for part (a) is asking since I can't imagine a ball rising once it is released...unless it is thrown upwards?

The entire balloon, with the ball in it, is rising at 50 ft per second so the initial velocity, C above, is 50 ft per second. That means dx/dt= 50- ct.
Initially the balls speed is 50 feet per second upward. When does dx/dt become 0 and then negative?

No, it does't. The problem is asking for a specific time, not a formula. The ball's initial velocity is 50 m/s upward. When will it be 0? How far will it rise in that time?

Part (b)
Assuming that the above equation for distance that I derived is correct, I simply needed to solve for t in x(t) = 0.
I used the quadratic formula and got t = 4.49s (ignoring the negative solution since time must be a positive value).

I would suggest that, since height and initial speed are given in the English system, you use g= -32.2 feet per second squared rather than change everything to metric system.

Differential calculus is the study of the rates at which quantities changes. The primary objects of study in differential calculus are the derivatives of a function, related notions such as the differential and their applications.

Differential calculus is the study of the rates at which quantities changes. The primary objects of study in differential calculus are the derivatives of a function, related notions such as the differential and their applications.